Question: Find the sum of the first $25$ terms in this geometric series: $8 + 6 + 4.5...$ Choose 1 answer: Choose 1 answer: (Choice A) A $ 0.03 $ (Choice B) B $4.57$ (Choice C) C $29.91$ (Choice D) D $ 31.98 $
Answer: Getting started We're dealing with a geometric series because each term is multiplied by $0.75$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {8})$ and the number of terms $(n = {25})$ are given in the question. The common ratio $r$ is ${0.75}$ because each term is multiplied by ${0.75}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{25}}&=\dfrac{{8}(1-\left({0.75}\right)^{{25}})}{1-\left({0.75}\right)} \\\\ S_{{25}}&=32(1-\left({0.75}\right)^{{25}})\\\\ S_{{{25}}} &\approx 31.98 \end{aligned}$ The answer $ 31.98 $